DOC. 25 FOUNDATIONS OF GENERAL
THEORY 285
8.
"Very
true,"
thinks the
reader,
"but the fact that Messrs. Einstein and
Grossmann
are
not
able
to
give
the
equations
for the
gravitational
field
in
generally
covariant form is
not
a
sufficient
reason
for
me
to
agree
to
a
specialization
of
the
reference
system."
But there
are
two
weighty arguments
that
justify
this
step, one
of
them
of
logical,
the other
one
of
empirical provenance:
a)
If
the
reference
system
is
chosen
totally arbitrarily,
then
the
guv
can by no [11]
means
be
completely
determined
by
the
2!ov.
For
imagine
that the
2!ov
and
gov
are
given everywhere,
and that all the
$ov
vanish in
one
part
&
of the four-dimensional
space.
I
can now
introduce
a new
reference
system
that
agrees completely
with the
original
one
outside
of O
but differs from it
(without
a
violation
of
continuity)
within
3.
If
one now
refers
everything
to
this
new
reference
system,
where the
matter is
represented by
$!ov.'
and the
gravitational
field
by
guv',
then,
even
though
we
do
have
IOV
'
=
IOV
everywhere,
the
equations
guv' =
guv
are
certainly
not
all satisfied in
the interior of
3.3
This
proves
the assertion.
Should
one
want to
make it
possible
for
the
guv (gravitational
field)
to
be
completely
determined from the
$ov(matter),
then this could
only
be achieved
by
restricting
the choice of the reference
system.
b)
In the
original theory
of
relativity,
the
momentum
and
energy
conservation law
[12]
is
expressed by
an
equation
of the form
£
=
0
(3)
v
oxv
The
corresponding equation
obtained with
the
help
of
the
absolute differential
calculus is
E
=
4
E
^
Y,.,
*•*
(4)
v
2^vt oxa
Equation
(4) no longer
has the form of
a pure
conservation law. This
is
understand-
able from
a
physical standpoint
insofar
as
the
matter,
considered
by
itself,
cannot
satisfy
the conservation laws in the
presence
of
a
gravitational field,
because the
gravitational
field transfers
momentum
and
energy
to the
matter.
This is
expressed
on
the
right-hand
side of
equation (4).
But
if
the conservation laws
are
to
remain
at
all
valid,
we
must
demand that there be conservation
laws of the form
of
(3)
for
the
3The
equations
are
to be understood in such
a way
that the
same
numerical values
are
always assigned
to the
independent
variables x'v
on
the left sides
as
to the variables
xv on
the
right
sides.
[13]